Abstract

In this thesis we focus on sets with a uniform rate of escape under the iteration of transcendental entire functions. We study their properties and their structure as well as using them as a tool to prove some interesting topological results.

First, motivated by the work of Rippon and Stallard, we generalise the quite fast escaping set by introducing a family of sets that escape to infinity at a uniform rate associated with the maximum modulus of the function. We examine under which conditions these sets are equal to the fast escaping set, which plays an important role in the iteration of transcendental entire functions. We prove that, in some cases, points which satisfy a relatively weak condition are actually fast escaping. We also show that this is not always the case by constructing an example of a function whose fast escaping set is not equal to these newly introduced sets.

Secondly, we look at the so-called spider's web structure and we give some general results which ensure that the escaping set, or a superset thereof, contains a spider's web. In particular, we prove that for a well-known function that was first studied by Fatou the escaping set has the structure of a spider's web. Using similar techniques, we also generalise this result to a class of functions all of which have one Baker domain.

Finally, motivated by the connection between spiders' webs and the non-escaping endpoints of Cantor bouquet Julia sets that first appeared when we studied Fatou's function, we present some topological results on the non-escaping endpoints of functions in the exponential family. More specifically, we show that the set of non-escaping endpoints together with infinity forms a totally separated set for every function in the exponential family whose singular value belongs to the Fatou set. This result is complementary to one of Alhabib and Rempe-Gillen who showed that, for the same class of functions, infinity is an explosion point for the set of escaping endpoints.

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